DOCSLIB.ORG

Some Topics in Extremal Combinatorics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Topics in Extremal Combinatorics Some topics in Extremal Combinatorics Hong Liu August 9, 2019 The purpose of this note is to give a touch on some topics in extremal combinatorics. These topics are chosen semi-randomly :) When possible, I try to present the proofs \back- wards", or with some intuitions here and there. The proofs would be a bit longer than usual, but hopefully they look more natural this way. The material covered are based on various notes/books/papers, see main texts for refer- ences. Comments are welcome, if you spot mistakes, please let me know :) 2 Contents 1 Extremal graph theory 5 1.1 Tur´antheorem . .5 1.2 Zykov's symmetrisation . .6 1.3 Erd}os-Stonetheorem . .7 1.4 Stability method . .8 2 Szemer´edi'sregularity lemma and its applications 11 2.1 Taster session . 11 2.2 Formal setup . 12 2.3 Key lemmas . 15 2.3.1 Reduced graph . 15 2.3.2 Embedding lemma . 16 2.3.3 Counting lemma . 16 2.4 Ruzsa-Szemer´editriangle removal lemma . 17 2.4.1 Cleaning the graph G ........................... 17 2.4.2 Proof of triangle removal lemma . 18 2.5 (6; 3)-theorem and Roth's theorem . 18 2.6 Ramsey-Tur´anproblem for K4 .......................... 20 2.7 Chv´atal-R¨odl-Szemer´edi-Trotter theorem . 21 2.8 Spectral proof of regularity lemma . 22 3 Pseudorandomness 27 3.1 Quasirandom graphs . 27 3.1.1 Equivalent definitions of quasirandomness . 28 3.1.2 (Codegree) ) (Induced Subgraph Count). 29 3.1.3 (4-cycle Count) ) strongly regular via Cauchy-Schwarz . 32 3 4 Chapter 1 Extremal graph theory In this chapter, we will discuss the classical Tur´an'stheorem in extremal graph theory and present some standard techniques such as Zykov's symmetrisation, stability method. 1.1 Tur´antheorem Before we start, let us consider the following puzzle. Suppose we have to choose n irrational numbers x1; : : : ; xn. How can we maximise the number of pairs (xi; xj) such that xi + xj is rational? One of the most classical extremal problems, nowadays so-called Tur´an-type problem, is: Problem 1.1.1 (Tur´an-type). How dense a graph can be without containing another (usually small) graph as a subgraph? More specifically, given a graph H, we say a graph G contains a copy of H, or H is a subgraph of G, or H ⊆ G, if there is an injective map ' : V (H) ! V (G) that preserves adjacencies, i.e. for any uv 2 E(H), we have '(u)'(v) 2 E(G). We call such a map an embedding of H in G. We say G is H-free if it does not contain H as a subgraph. If in addition, the map preserves also non-adjacencies, then H is an induced subgraph of G. The main parameter we study for Problem 1.1.1 is the extremal number of H, ex(n; H) = maxfe(G): jGj = n and G is H-freeg; is the maximum size of an n-vertex H-free graph. We call an n-vertex graph G an extremal graph for H, if G is H-free of maximum size, i.e. e(G) = ex(n; H). One of the earliest applications of extremal graph theory, by Erd}os,is to construct dense multiplicative Sidon set of integers using a graph without 4-cycles. The first result in extremal graph theory is the following theorem of Mantel, which answers Problem 1.1.1 when forbidding triangles as subgraphs. Theorem 1.1.2 (Mantel 1907). Let G be an n-vertex graph. If G is triangle-free, then 2 e(G) ≤ ex(n; K3) = bn =4c: 5 Exercise 1.1.3. Solve the puzzle at the beginning of this section, i.e. find the maximum 1 number of pairs of irrationals (xi; xj) with xi + xj being rational. Exercise 1.1.4. Prove that for any tree T , ex(n; T ) = O(n), that is, there exists a constant C = C(T ) such that ex(n; T ) ≤ Cn.2 Mantel's result in fact shows that extremal graph for triangle is Kbn=2c;dn=2e. This answers also the following natural question for triangles. Problem 1.1.5 (Extremal structure/Stability). How do H-extremal graphs look like? What about almost extremal graphs3, do they look like extremal ones? Theorem 1.1.2 was later generalised by Tur´anto forbidding larger cliques. To state his result, we need to define a special family of graphs. Let r 2 N, the r-partite Tur´angraph on n vertices, denoted by Tr(n), is the balanced complete r-partite n-vertex graph, i.e. each partite set is of size either bn=rc or dn=re. Clearly, Tr(n) is Kr+1-free. Theorem 1.1.6 (Tur´an1941). Let r ≥ 2 be an integer and G be an n-vertex graph. If G is Kr+1-free, then 1 n2 e(G) ≤ ex(n; K ) = e(T (n)) = 1 − − O(r): r+1 r r 2 Furthermore, the Tur´angraph Tr(n) is the unique extremal graph. We see from Tur´antheorem that there is a unique extremal graph Tr(n). The following theorem of Erd}osand Simonovits shows that this problem is stable in the sense that every almost extremal graph must be close in structure to the extremal Tur´angraph, answering Problem 1.1.5 for cliques. Theorem 1.1.7 (Erd}os-Simonovits stability 1966). Let " > 0, there exists δ > 0 such that the following holds. Let G be an n-vertex Kr+1-free graph. If 2 e(G) ≥ ex(n; Kr+1) − δn ; 2 then G can be changed to Tr(n) by altering at most "n adjacencies. 1.2 Zykov's symmetrisation There are many proofs for Tur´antheorem. Here we present one using Zykov's symmetri- sation. Zykov's symmetrisation is a process in which we alter the graph, one vertex at a time, 1Hint: Build an auxiliary graph and apply Mantel's theorem. 2Prove first that every graph with average degree d contains a subgraph with minimum degree at least d=2. 3We say G is almost extremal for H if G is H-free and close to maximum size, i.e. e(G) ≥ ex(n; H)−o(n2). 6 • without decreasing the number of edges, and • without increasing the clique number !(G).4 At the end of the process, we arrive to a complete partite graph, which has a much simpler structure to deal with. In particular, if the original graph is Kr+1-free, then all the graphs during symmetrisation will be Kr+1-free. Proof of Tur´antheorem via Zykov's Symmetrisation. Let G be an n-vertex Kr+1-extremal graph. Pick v1 2 V (G) with maximum degree and symmetrise all of its non-neighbours to v1. That is, for each u not adjacent to v1, set N(u) := N(v1). This operation keeps Kr+1-freeness and the resulting graph G1 has at least as many edges as G. Note that in G1, V1 := V n N(v1) is an independent set and completely joined to N(v1). We now repeat this operation as follows. Pick v2 2 G1[N(v1)] and symmetrise all its non- neighbours to v2. Let G2 be the resulting graph, then again G2 is Kr+1-free and e(G2) ≥ e(G1) ≥ e(G). Note that in G2, V2 := V (G2) n N(v2) is an independent set and completely joined to N(v2). 0 Continue this process, we will get a complete partite graph, say G , that is also Kr+1- free at the end. As the original graph G is an extremal Kr+1-free graph, together with e(G0) ≥ e(G), we see that G0 must be also extremal, i.e. e(G0) = e(G). We leave the uniqueness of extremal graph as exercise. Exercise 1.2.1. Among all Kr+1-free complete partite graphs, the Tur´angraph Tr(n) is the unique extremal graph. Further readings. Symmetrisation trick has been used in various extremal problems. To begin, one can read the linear algebraic version, and a recent generalisation due to F¨uredi and Maleki that can be applied to multiple graphs simultaneously. See also Pikhurko-Staden- Yilma for another application on Erd}os-Rothschild problem. • Motzkin and Straus, Maxima for graphs and a new proof of a theorem of Tur´an, Canad. J. Math., (1965). • F¨urediand Maleki, The minimum number of triangular edges and a symmetrization method for multiple graphs, Combin. Probab. Comput., (2017). • Pikhurko, Staden, and Yilma, The Erd}os-Rothschild problem on edge-colourings with forbidden monochromatic cliques, Math. Proc. Cambridge Phil. Soc. (2017). 1.3 Erd}os-Stonetheorem We have seen that Tur´antheorem determines the extremal number for cliques and describes the unique extremal structure. The natural next step is what if we forbid general graphs 4The clique number !(G) of a graph G is the order of the largest clique contained in G. 7 other than cliques? We shall present in this section a satisfying answer for all non-bipartite graphs. The seminal result of Erd}osand Stone shows that the extremal function of a general graph is completely determined by another important graph parameter: the chromatic number. Recall that the chromatic number of a graph H, denoted by χ(H), is the minimum number of colours needed to colour V (H) so that adjacent vertices do not receive the same colour. Theorem 1.3.1 (Erd}os-Stone1946). Let H be an arbitrary graph, then5 1 n2 ex(n; H) = 1 − + o(1) : χ(H) − 1 2 Note that by the definition of chromatic number, the (χ(H) − 1)-partite Tur´angraph is H-free, yielding the lower bound above.
Load more

Recommended publications
  • Graph Theory
    Graph theory Eric Shen Friday, August 28, 2020 The contents of this handout are based on various lectures on graph theory I have heard over the years, including some from MOP and some from Canada/USA Mathcamp. Contents 1. Basic terminology3 1.1. Introduction to graphs ................................ 3 1.2. Dictionary ....................................... 3 2. Basic graph facts4 2.1. Trees .......................................... 4 2.2. Handshaking lemma.................................. 5 2.3. Example problems................................... 5 3. Colorings of graphs7 3.1. Introduction to coloring................................ 7 3.2. Planar graphs ..................................... 7 3.3. Four-color theorem .................................. 8 3.4. Example problems................................... 8 4. Combinatorial constructions 10 4.1. Basic graph constructions............................... 10 4.2. Perfect matchings................................... 11 4.3. Corollaries of Hall's theorem............................. 11 4.4. A hard walkthrough.................................. 12 5. Extremal graph theory 13 5.1. Triangle-free bounds.................................. 13 5.2. Formalization ..................................... 14 5.3. Tur´angraphs...................................... 14 5.4. Example problems................................... 15 6. Application: crossing numbers 17 6.1. Conventional definitions................................ 17 6.2. Some conjectures ................................... 17 6.3. Crossing number
    [Show full text]
  • The Chromatic Number of Infinite Graphs —
    Discrete Mathematics 311 (2011) 1448–1450 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc The chromatic number of infinite graphs — A survey Péter Komjáth Department of Computer Science, Eötvös University, Budapest, P.O. Box 120, 1518, Hungary article info a b s t r a c t Article history: We survey some old and new results on the chromatic number of infinite graphs. Available online 26 November 2010 ' 2010 Elsevier B.V. All rights reserved. Keywords: Infinite graphs Chromatic number 1. Coloring number The coloring number of a graph X; Col.X/, is the smallest cardinal µ such that V , the set of vertices of X, has a well- ordering < such that each vertex is joined into less than µ smaller (by <) vertices. This notion was introduced by Erd®s and Hajnal in [2]. If Col.X/ D κ and < is a well-ordering witnessing it, then a straightforward transfinite recursion via < gives a good coloring of X with κ colors; at each step we can continue the coloring. This establishes Chr.X/ ≤ Col.X/ for any graph C X. Equality may not hold even for very simple graphs as Col.Kκ,κ / D κ and Kκ,λ D κ for κ < λ. Here Kκ,λ denotes the complete bipartite graph with bipartition classes of size κ; λ, respectively. What makes the notion of coloring number particularly useful is Shelah's Singular Cardinals Compactness Theorem: if λ > µ, λ is singular, X is a graph such that Col.Y / ≤ µ holds for every subgraph Y of X with jY j < λ, then Col.X/ ≤ µ holds, as well [22].
    [Show full text]
  • Bounds for Graph Regularity and Removal Lemmas
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace@MIT Bounds for graph regularity and removal lemmas David Conlon∗ Jacob Foxy Abstract We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck2= log∗ k pairs of parts which are not -regular, where c; > 0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer´edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter may require as many as −2 2Ω( ) parts. This is tight up to the implied constant and solves a problem studied by Lov´asz and Szegedy. 1 Introduction Originally developed by Szemer´edias part of his proof of the celebrated Erd}os-Tur´anconjecture on long arithmetic progressions in dense subsets of the integers [39], Szemer´edi'sregularity lemma [40] has become a central tool in extremal combinatorics.
    [Show full text]
  • Large Networks and Graph Limits László Lovász
    Large networks and graph limits L´aszl´oLov´asz Institute of Mathematics, Eotv¨ os¨ Lorand´ University, Budapest, Hungary 2010 Mathematics Subject Classification. Primary 05C99, Secondary 05C25, 05C35, 05C80, 05C82, 05C85, 90B15 Key words and phrases. graph homomorphism, graph algebra, graph limit, graphon, graphing, property testing, regularity lemma To Kati as all my books Contents Preface xi Part 1. Large graphs: an informal introduction 1 Chapter 1. Very large networks 3 1.1. Huge networks everywhere 3 1.2. What to ask about them? 4 1.3. How to obtain information about them? 5 1.4. How to model them? 8 1.5. How to approximate them? 11 1.6. How to run algorithms on them? 18 1.7. Bounded degree graphs 22 Chapter 2. Large graphs in mathematics and physics 25 2.1. Extremal graph theory 25 2.2. Statistical physics 32 Part 2. The algebra of graph homomorphisms 35 Chapter 3. Notation and terminology 37 3.1. Basic notation 37 3.2. Graph theory 38 3.3. Operations on graphs 39 Chapter 4. Graph parameters and connection matrices 41 4.1. Graph parameters and graph properties 41 4.2. Connection matrices 42 4.3. Finite connection rank 45 Chapter 5. Graph homomorphisms 55 5.1. Existence of homomorphisms 55 5.2. Homomorphism numbers 56 5.3. What hom functions can express 62 5.4. Homomorphism and isomorphism 68 5.5. Independence of homomorphism functions 72 5.6. Characterizing homomorphism numbers 75 5.7. The structure of the homomorphism set 79 Chapter 6. Graph algebras and homomorphism functions 83 6.1.
    [Show full text]
  • Regular Partitions of Gentle Graphs
    REGULAR PARTITIONS OF GENTLE GRAPHS YITING JIANG, JAROSLAV NESETˇ RIL,ˇ PATRICE OSSONA DE MENDEZ, AND SEBASTIAN SIEBERTZ Dedicated to Endre Szemer´edion the occasion of his eightieth birthday. Abstract. Szemer´edi'sRegularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as \gentle" classes. Contents Introduction2 Part 1. Preliminaries and survey of some regularity lemmas 4 1. Model theory background4 2. Graph theoretic background5 3. Random-free regularity lemmas9 Part 2. Regularity for gentle graphs 15 4. Set-defined classes: both stable and semi-algebraic regularity 15 5. Order-defined classes 18 6. Inherited regularity of 2-covered classes 19 7. Regularity for classes 2-covered by embedded m-partite cographs 22 8. Regularity and non-regularity for nowhere dense classes 26 Conclusion 29 arXiv:2003.11692v2 [math.CO] 29 Mar 2020 Acknowledgments 30 References 30 This paper is part of a project that has received funding from the European Research Council (ERC) under the Euro- pean Union's Horizon 2020 research and innovation programme (grant agreement No 810115 { Dynasnet). 1 REGULAR PARTITIONS OF GENTLE GRAPHS 2 Introduction Szemer´edi'sRegularity Lemma. Szemer´edi'sRegularity Lemma [77] is a very useful tool in extremal graph theory.
    [Show full text]
  • Graph Theory and Additive Combinatorics, a Graduate-Level Course Taught by Prof
    GRAPHTHEORY AND ADDITIVECOMBINATORICS notes for mit 18.217 (fall 2019) lecturer: yufei zhao http://yufeizhao.com/gtac/ About this document This document contains the course notes for Graph Theory and Additive Combinatorics, a graduate-level course taught by Prof. Yufei Zhao at MIT in Fall 2019. The notes were written by the students of the class based on the lectures, and edited with the help of the professor. The notes have not been thoroughly checked for accuracy, espe- cially attributions of results. They are intended to serve as study resources and not as a substitute for professionally prepared publica- tions. We apologize for any inadvertent inaccuracies or misrepresen- tations. More information about the course, including problem sets and lecture videos (to appear), can be found on the course website: http://yufeizhao.com/gtac/ Contents A guide to editing this document 7 1 Introduction 13 1.1 Schur’s theorem........................ 13 1.2 Highlights from additive combinatorics.......... 15 1.3 What’s next?.......................... 18 I Graph theory 21 2 Forbidding a subgraph 23 2.1 Mantel’s theorem: forbidding a triangle.......... 23 2.2 Turán’s theorem: forbidding a clique............ 24 2.3 Hypergraph Turán problem................. 26 2.4 Erd˝os–Stone–Simonovits theorem (statement): forbidding a general subgraph...................... 27 2.5 K˝ovári–Sós–Turán theorem: forbidding a complete bipar- tite graph............................ 28 2.6 Lower bounds: randomized constructions......... 31 2.7 Lower bounds: algebraic constructions.......... 34 2.8 Lower bounds: randomized algebraic constructions... 37 2.9 Forbidding a sparse bipartite graph............ 40 3 Szemerédi’s regularity lemma 49 3.1 Statement and proof....................
    [Show full text]
  • [Cs.DM] 12 Aug 2021 Agnostic Online Learning and Excellent Sets
    Agnostic Online Learning and Excellent Sets M. Malliaris∗ and S. Moran† September 6, 2021 Abstract We use algorithmic methods from online learning to revisit a key idea from the interaction of model theory and combinatorics, the existence of large “indivisible” sets, called “ǫ-excellent,” in k-edge stable graphs (equivalently, Littlestone classes). These sets arise in the Stable Regularity Lemma, a theorem characterizing the appearance of irregular pairs in Szemer´edi’s celebrated Regularity Lemma. Translating to the language of probability, we find a quite different existence proof for ǫ-excellent sets in Littlestone classes, using regret bounds in online learning. This proof k applies to any ǫ < 1/2, compared to < 1/22 or so in the original proof. We include a second proof using closure properties and the VC theorem, with other advantages but weaker bounds. As a simple corollary, the Littlestone dimension remains finite under some natural modifications to the definition. A theme in these proofs is the interaction of two abstract notions of majority, arising from measure, and from rank or dimension; we prove that these densely often coincide and that this is characteristic of Littlestone (stable) classes. The last section lists several open problems. 1 Overview In this section we briefly present three complementary points of view (combinatorics §1.1, online learning §1.2, model theory §1.3) which inform this work, and state our main results in §1.4. The aim is to allow the paper to be readable by people in all three communities and hopefully to stimulate further interaction. In the recent papers [4], [7] ideas from model theory played a role in the conjecture, and then arXiv:2108.05569v2 [cs.DM] 3 Sep 2021 the proof, that Littlestone classes are precisely those which can be PAC learned in a differentially private way (we direct the reader to the introductions of those papers for precise statements and further literature review).
    [Show full text]
  • Some New and Old Problems on Chromatic Graphs
    SOME NEW AND OLD PROBLEMS ON CHROMATIC GRAPHS By PAUL EIRDÖS Hungarian Academy of Scie3 ces This note should be considered as a preliminary announcement, some of the references are incomplete and might even be inaccurate . The reason is that some of the new results are not yet in their final form and since many of them are only a few weeks old I may not be able to give the exact authorship . All these possible errors will soon be corrected in later publications, but enough of these apologies and let us start the mathematics . Let G be a graph of chromatic number n . Rodl defines : G is said to have local chromatic number y (r < n) if r is the smallest integer for which there is an integer p (p > n) so that G has a good colouring by p colours (a colouring is called good if two vertices which are joined always get different colours) and the star of every vertex gets fewer than r colours i.e . the vertex together with its star gets at most r colours . First of all it is not at all obvious that such graphs exist. But Rödl observed that an old graph of Hajnal and myself [1] has chromatic number n (for every finite and infinite n) and local chromatic number 3 . By the way it is easy to see that if the local chromatic number is 2 then the graph must be bipartite . Let the vertices of G be the ordered triples (i, j, k), 1 < i < j < k < N, (i, j, k) is joined to (u, v, w) if and only if u =J, v = k .
    [Show full text]
  • Some Extremal and Structural Problems in Graph Theory Taylor Mitchell Short University of South Carolina
    University of South Carolina Scholar Commons Theses and Dissertations 2016 Some Extremal And Structural Problems In Graph Theory Taylor Mitchell Short University of South Carolina Follow this and additional works at: https://scholarcommons.sc.edu/etd Part of the Mathematics Commons Recommended Citation Short, T. M.(2016). Some Extremal And Structural Problems In Graph Theory. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/3753 This Open Access Dissertation is brought to you by Scholar Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Some extremal and structural problems in graph theory by Taylor Mitchell Short Bachelor of Science College of William and Mary 2008 Master of Science Virginia Commonwealth University 2011 Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics College of Arts and Sciences University of South Carolina 2016 Accepted by: László Székely, Major Professor Éva Czabarka, Committee Member Linyuan Lu, Committee Member Csilla Farkas, Committee Member Lacy Ford, Senior Vice Provost and Dean of Graduate Studies c Copyright by Taylor Mitchell Short, 2016 All Rights Reserved. ii Acknowledgments First and foremost, thanks to my advisor László Székely. I have learned so much from not only the lessons you taught me, but also from simply being around you. Thank you for your patience, support, and inspiration. Thank you to Éva Czabarka. Your advice has been invaluable. Working closely with you has been an absolute pleasure. To Tomislav Došlić, thank you for hosting me during the Summer 2015.
    [Show full text]
  • Wiener Index of Chain Graphs SHAHISTHA, K ARATHI BHAT and SUDHAKARA G
    IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_10 ______________________________________________________________________________________ Wiener Index of Chain Graphs SHAHISTHA, K ARATHI BHAT and SUDHAKARA G Abstract—A bipartite graph is called a chain graph if the concepts and notations. neighborhoods of the vertices in each partite set form a chain The distance d(u; v) between a pair of vertices u; v in G is with respect to set inclusion. Chain graphs are discovered the length of the shortest path between u and v if exists, and re-discovered by various researchers in various contexts. Also, they were named differently according to the applications else d(u; v) = 1. The Wiener index is a graph invariant in which they arise. In the field of Spectral Graph Theory, which belongs to the molecular structure descriptors known chain graphs play a remarkable role. They are characterized as topological indices, which are used for the designing the as graphs with the largest spectral radius among all the molecules with required properties according to the practical connected bipartite graphs with prescribed number of edges convinience. The Wiener index w(G) of a graph G is the and vertices. Even though chain graphs are significant in the field of Spectral Graph Theory, the area of graph parameters sum of all distances between all pairs of vertices in G. remains untouched. Wiener index is one of the oldest and most X studied topological indices, both from theoretical point of view w(G) = d(u; v): and applications. The Wiener index of a graph is defined as fu;vg2V (G) the sum of distances between every pair of vertices in it.
    [Show full text]
  • Abstract Szemerédi's Regularity Lemma Is a Deep Result in Graph Theory
    Abstract Szemer´edi’s regularity lemma is a deep result in graph theory with applications in many different areas of mathematics. The lemma says that any graph can be approximated by the union of a bounded num- ber of random-like bipartite graphs and this can be used to extract the underlying structure of the graph. Recently it has been shown that there exists polynomial time algorithms that can make this ap- proximation. This survey gives a proof of the regularity lemma, shows some applications and discusses some algorithmic aspects. Contents 1 Introduction 1 2 Notations 2 3 The Regularity Lemma 3 3.1 Definitions............................. 3 3.2 Szemer´edi’s Regularity Lemma . 4 3.3 Proofoftheregularitylemma . 4 3.4 Bounds .............................. 10 4 Historical background 12 4.1 vanderWaerden’stheorem . 12 4.2 Szemer´edi’stheorem . 12 4.3 Theoriginallemma ........................ 14 5 An extremal problem 15 5.1 Theregularitygraph . 15 5.2 Erd¨os-Stonetheorem . 18 6 3-term arithmetic progressions 21 6.1 Induced matching problem . 21 6.2 The (6, 3)problem ........................ 22 6.3 3-termarithmeticprogressions . 24 7 Algorithmic aspects of the regularity lemma 26 7.1 Testingforregularity . 26 7.2 Creatingtheregularitypartitions . 27 7.3 Algorithmicapplications . 29 8 Extending the regularity lemma 30 8.1 The regularity lemma for sparse graphs . 30 8.2 The hypergraph version of the regularity lemma . 31 9 Recent results using the regularity lemma 33 9.1 Uniform edge distribution and k-universalgraphs . 33 9.2 Theblow-uplemma........................ 36 A Graphs and hypergraphs 38 B Extremal graph theory 38 i C Ramsey Theory 40 D NP-completeness 40 ii 1 Introduction In 1936 the famous mathematicians Paul Erd¨os and Paul Tur´an conjectured 1 that if A = a1, a2,..
    [Show full text]
  • Contents 1. Introduction 1 2
    STABLE GRAPHS BENJAMIN OYE Abstract. In Regularity Lemmas for Stable Graphs [1] Malliaris and She- lah apply tools from model theory to obtain stronger forms of Ramsey's theo- rem and Szemeredi's regularity lemma for stable graphs," graphs which admit a uniform nite bound on the size of an induced sub-half-graph. This paper provides a background to the rst theorem of that , an improved form of Ramsey's theorem for stable graphs without model theory as a prerequisite. Contents 1. Introduction 1 2. Denitions 2 3. Order and Indiscernibles 3 4. Random Graphs 3 5. Orderly graphs 4 6. Stable Simple Graphs 5 7. Stable Hypergraphs 9 Acknowledgments 12 References 12 1. Introduction Ramsey theory studies the conditions under which order arises in mathematical structures. While Ramsey problems are often stated in the language of graphs, the connection to logic has been evident since F.P Ramsey's 1930 On a Problem of Formal Logic which proved the central theorem now known as Ramsey's theorem. The theorem which will be developed in this paper was proved in a paper ul- timately aimed at improving Szemerédi's Regularity Lemma for a certain class of graphs. Theorem 1.1 (Szemerédi's Regularity Lemma). For every , m there exist N = N(, m), m0 = m(, m) such that given any nite graph X of at least N vertices, 0 there is a k with m ≤ k ≤ m and a partition X = X1 [···[ Xk satisfying: (1) jjXij − jXjjj ≤ 1 for all i; j ≤ k and 2 (2) all but at most k of the pairs (Xi;Xj) are -regular.
    [Show full text]